Limiting Spectral Density Of A Class Of Special Sample Covariance Matrices

In recent decades,with the rapid development of technology and science the rapid growth of computer computing speed and storage has made it possible to analyze the large dimensional data.However,many classical statistical methods are no longer applicable when we are dealing with large dimensional data,and some of them even have serious errors.Large dimensional random matrix theory has attracted wide attention of statistician because it can deal with large dimensional data effectively.There are many research questions which can be handled by large dimensional random matrix knowledge,such as multivariate regression analysis,signal processing,etc.In addition,many test statistics constructed for hypothesis test problems are related to the sample covariance matrix,which is the most common kind of random matrix,so it is very important to study the spectral properties of sample covariance matrix.Mar(?)enko and Pastur(1967)proved that the empirical spectral distribution of sample covariance matrix converges to M-P law in probability.On the basis of the work,we apply it to study the limiting spectral density function of sample covariance matrix when the population covariance matrix has multiclass of nonzero population eigenvalues.In this paper,according to the spectral analysis knowledge of large dimensional sample covariance matrix,we give a set of detailed derivation process about the proposed model of the limiting spectral density function of sample covariance matrix,and provide the algorithm about how to determine the support set of limiting spectral distribution function and make the graph of the limiting spectral density function.In addition,some numerical results are obtained by programming to show the operability and effectiveness of this method.